� Incremental method + co –rotationnal FE formulation + contact penalty-barrier method

lamcos.insa-lyon.frLaMCoS, Université de Lyon, CNRS, INSA-Lyon UMR5259, 18-20 rue des Sciences - F69621 Villeurbanne Cedex

Research steps Drag force and Efficiency analysis

Technological and Scientific contexts

� Drag force: threshold load to move the cable into the casing

� Efficiency: ratio between In/Out forces

� Experimental design influence analysis � Drag and Efficiency per unit length

Nonlinear interactions between a cable with no axial preload and its casing. Application to automotive gearshift command systems

N. LEVECQUE 1,2, Dr. L. MANIN1, Dr. T. NOURI BARANGER1, Pr. R. DUFOUR1

1LaMCoS INSA-Lyon, 2 RENAULT S.A

Dynamic and Vibration transmission analysis

Curvilinear casing geometry and contact points prediction

casing casing casing casing stopstopstopstop

casing stopcasing stopcasing stopcasing stop

rubber rubber rubber rubber guideguideguideguide

available available available available spacespacespacespace

car floorcar floorcar floorcar floor

Gearshift Gearshift Gearshift Gearshift boxboxboxbox

gearboxgearboxgearboxgearbox

rubber passrubber passrubber passrubber pass----throughthroughthroughthrough

cable

casing

leverleverleverlever

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 2 4 6 8 10

Curvature (m-1)

Effi

cien

cy F_o

ut/F

_in

pushpull

0

2

4

6

8

10

12

14

0 2 4 6 8 10curvature (m-1)

Dra

g fo

rce

per

unit

leng

th (

N/m

)

pushpull

Driving comfort requires to reduce:

- Drag friction force during gear shifting,

- Gearbox vibration transmission to the gearshift lever

- cable-casing vibro-impacting noise

Main scientific challenges

- cable-casing spatially curved geometry simulation,

- non-linearities: cable-casing contact, large displacement,

- vibrations and dynamic instabilities, parametric excitations

5 2 2 6 2 70,559 0,006. 39,11. 0,001. 2,38.10 . 520,4. 3,1.10 . 0,06. . 6,6.10 . . 0,019. .push F C T F C T F C F T C T− − −− − −− − −− − −= + − + − − − − + += + − + − − − − + += + − + − − − − + += + − + − − − − + +ρρρρ

2 -4 24,41-1251,6. - 0,038. 327950. 5,4.10 . -7,4. .pushDrag C T C T C T= + +2 -4 23,957 -336,8. - 0,087. 221783. 10,1.10 . - 8,09. .pullDrag C T C T C T= + += + += + += + +

5 2 2 6 2 70,693 0,003. 27,03. 0,002. 1,03.10 . 3083,6. 11,3.10 0,02. . 44,5.10 . . 0,079. .pull F C T F C T F C F T C Tρρρρ − − −− − −− − −− − −= + − + − − − − + += + − + − − − − + += + − + − − − − + += + − + − − − − + +

Cable and casing mechanical properties (E,

EI, ρρρρ), geometry characteristics

Boundary conditions and surrounding components

(start, end, intermediate, points)

Non linear FE Computation, large displacements, co-

rotational formulation, cable-casing contact problem

solved with penalty method

Elementary Drag force and

efficiency experimental

identification

Drag force and efficiency calculation on real

car gearshift mechanism

Dynamic

Ananysis

• Modes and frequencies computation

around static equilibrium position

• Forced response to:

• Cable axial excitation

• Casing transverse excitation

• Vibration instabilities prediction

Spatially curved cable-casing path,

contact points and forces

Drag force and efficiency

models integration

Experimental validation

kEkEk

k EE k

E

kE

kE

u

v

wU

x

y

z

θθθ

=

kO

kO

kk OO k

OkO

kO

u

v

wU

x

y

z

θθθ

=

18 50 100 150 200

[Hz]

0.001

0.01

0.1

[g/(m/s²)]1. 002:

2. 002:

3. 002:

4. 002:

12

34

18 50 100 150 200

[Hz]

0.001

0.01

0.1

[g/(m/s²)]1. 002:

2. 002:

3. 002:

4. 002:

1

23

4

k = 0

( )( )

int

int

k ktc c C C C

k ktG G G G G

K U F R U

K U F R U

= −

= −

,

,

k kC G

C extC G extG

P P

F F F F= =

( ),k kcont C GF F U U=

C extC cont

G extG cont

F F F

F F F

= += −

1

1

k k kC C C

k k kG G G

P P U

P P U

+

+

= += +

1

1

finale kC C

finale kG G

P P

P P

+

+

==

k = N-1

k < N-1

k = k+1

Newton iterativeconvergence loop

Incrementalloop

N2vN1

wuwu

v

θy

••

θx

θz

θyθx

θz

casingcvjeu

gv

cableδ

Planar-S casing path

FE modeling, cable-casing contact interaction

¼ circle casing path

Car configuration testCable axial excitation3-axes accelerometers

Input axial acceleration on cable

0

10

20

30

40

50

60

70

10 50 90 130 170 210Hz

Acc

el m

/s²

rms

Computation algorithm

•

••

•• •

iC

1iG +

1iG −

1iC +

1iC −

iG

nr

droitenr

gauchenr

•M

•

••

•• •

iC

1iG +

1iG −

1iC +

1iC −

iG

nr

droitenr

gauchenr

•M

Oi (xOi, yOi, zOi)

Ni+1 (xi+1, yi+1, zi+1)

Ni (xi, yi, zi)

Ni-1 (xi-1, yi-1, zi-1)

Ri

.n ig jeu C M n= −uuuur r

.

. 0

. 0n n

cont n n n na g

cont n n

F t e g si g

F t e si g

= + < = ≥

sin /

/ sin

.

.

r r

r rca g cable cont

cable ca g cont

F F n

F F n

= − =

Electricjack

Load anddisplacement

sensors

Powderbrake

FinFou

tFin

Fout

Fin

Fout

85°C

150 N40°C6.25 m-1

100 N0°C5 m-1

50 N-30°C∞

Force FTemperature TCurvature C

85°C

150 N40°C6.25 m-1

100 N0°C5 m-1

50 N-30°C∞

Force FTemperature TCurvature C

85°C

150 N40°C6.25 m-1

100 N0°C5 m-1

50 N-30°C∞

Force FTemperature TCurvature C

γγγγin

NEU

1NOU −

NOU

E

O

E

O Initial state

Final state

1OU

2OU3OU

1EU

OU

EU

D1 =0.378 m

D2 =0.278 m

D1 =0.378 m

D2 =0.278 m

O

E

UE

0 0.1 0.2 0.3 0.4 0.5

-2

0

2

x 10-4

curvilinear abscissa (m)

cabl

e-ca

sing

neu

tral

fibe

r di

stan

ce (

m)

0 0.1 0.2 0.3 0.4 0.5

0

0.02

0.04

0.06

0.08

cont

act f

orce

s m

odul

us (

N)

0 0.1 0.2 0.3 0.40

0.05

0.1

0.15

0.2

0.25

0.3

x (m)

y (

m)

computed cable-casing geometry

casingcable

_

_1

( ).i Nb z

lin i zone ii

Drag Drag R L=

=

= ∑

1 1out in out totalglobal

in in in

F F F F

F F Fη − ∆= = − = −

1

_1

n

total zone ii

F F−

=

∆ = ∆∑

_ _ _.zone i lin i zone iF F L∆ = ∆

__ 1 lin i

lin ii

F

Fη

∆= −

( )1__ −=∆ iliniilin FF η

(N/m)(N/m)

N3

N1 N2

Nn-2

Nn-1 Nn

Zone 1

E

Zone 2

Zone n-2

Zone n-1

R

3F

n-1F

n-2F

2FFin

Fout

� Drag force an efficiency computation from cable-casin g path decomposition andexperimentally identified model

(Fin,C1) ηlin_1 _1zoneF∆ 2 _1in zoneF F F= − ∆ ….

�Time response around static equilibrium position� cable axial harmonic excitation � casing transverse excitation

�Time integration to predict dynamic response�Experimentation-Validation on

� basic casing curved geometry� car cable-casing geometry

Computed cable-casingspatially curved geometry

Kcable-casing , Mcable-casing

Newmark schemeTime response computation

u(t)P(t)=P0 + u(t)

F(t)=F1sin(ωt)Or

γ(t)=γ1sin(ωt)

0 1 2 3 4 5

x 10-4

-0.03

-0.025

-0.02

-0.015

-0.01

-0.005

0

Cable-Casing neutral fiber distance (m)

Con

tact

For

ce (

N)

tn -0.001 N, en 10 N/mtn -0.001 N, en100 N/mtn -0.001 N, en 50 N/mtn -1e-4 N, en 10 N/mtn -1e-4 N, en 50 N/mtn -1e-5 N, en 10 N/mtn -0.00316 N, en 100 N/mtn -0.00316 N, en 50 N/mtn -0.005 N, en 100 N/m